9,520 research outputs found
Piecewise rigid curve deformation via a Finsler steepest descent
This paper introduces a novel steepest descent flow in Banach spaces. This
extends previous works on generalized gradient descent, notably the work of
Charpiat et al., to the setting of Finsler metrics. Such a generalized gradient
allows one to take into account a prior on deformations (e.g., piecewise rigid)
in order to favor some specific evolutions. We define a Finsler gradient
descent method to minimize a functional defined on a Banach space and we prove
a convergence theorem for such a method. In particular, we show that the use of
non-Hilbertian norms on Banach spaces is useful to study non-convex
optimization problems where the geometry of the space might play a crucial role
to avoid poor local minima. We show some applications to the curve matching
problem. In particular, we characterize piecewise rigid deformations on the
space of curves and we study several models to perform piecewise rigid
evolution of curves
Affine symmetry in mechanics of collective and internal modes. Part I. Classical models
Discussed is a model of collective and internal degrees of freedom with
kinematics based on affine group and its subgroups. The main novelty in
comparison with the previous attempts of this kind is that it is not only
kinematics but also dynamics that is affinely-invariant. The relationship with
the dynamics of integrable one-dimensional lattices is discussed. It is shown
that affinely-invariant geodetic models may encode the dynamics of something
like elastic vibrations
Gradient flows as a selection procedure for equilibria of nonconvex energies
For atomistic material models, global minimization gives the wrong qualitative behavior; a theory of equilibrium solutions needs to be defined in different terms. In this paper, a concept based on gradient flow evolutions, to describe local minimization for simple atomistic models based on the Lennard–Jones potential, is presented. As an application of this technique, it is shown that an atomistic gradient flow evolution converges to a gradient flow of a continuum energy as the spacing between the atoms tends to zero. In addition, the convergence of the resulting equilibria is investigated in the case of elastic deformation and a simple damaged state
An isogeometric finite element formulation for phase transitions on deforming surfaces
This paper presents a general theory and isogeometric finite element
implementation for studying mass conserving phase transitions on deforming
surfaces. The mathematical problem is governed by two coupled fourth-order
nonlinear partial differential equations (PDEs) that live on an evolving
two-dimensional manifold. For the phase transitions, the PDE is the
Cahn-Hilliard equation for curved surfaces, which can be derived from surface
mass balance in the framework of irreversible thermodynamics. For the surface
deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation.
Both PDEs can be efficiently discretized using -continuous interpolations
without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured
spline spaces with pointwise -continuity are utilized for these
interpolations. The resulting finite element formulation is discretized in time
by the generalized- scheme with adaptive time-stepping, and it is fully
linearized within a monolithic Newton-Raphson approach. A curvilinear surface
parameterization is used throughout the formulation to admit general surface
shapes and deformations. The behavior of the coupled system is illustrated by
several numerical examples exhibiting phase transitions on deforming spheres,
tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to
the text, added supplementary movie file
Mechanics of Systems of Affine Bodies. Geometric Foundations and Applications in Dynamics of Structured Media
In the present paper we investigate the mechanics of systems of
affinely-rigid bodies, i.e., bodies rigid in the sense of affine geometry.
Certain physical applications are possible in modelling of molecular crystals,
granular media, and other physical objects. Particularly interesting are
dynamical models invariant under the group underlying geometry of degrees of
freedom. In contrary to the single body case there exist nontrivial potentials
invariant under this group (left and right acting). The concept of relative
(mutual) deformation tensors of pairs of affine bodies is discussed. Scalar
invariants built of such tensors are constructed. There is an essential novelty
in comparison to deformation scalars of single affine bodies, i.e., there exist
affinely-invariant scalars of mutual deformations. Hence, the hierarchy of
interaction models according to their invariance group, from Euclidean to
affine ones, can be considered.Comment: 50 pages, 4 figure
Gravity duals for the Coulomb branch of marginally deformed N=4 Yang-Mills
Supergravity backgrounds dual to a class of exactly marginal deformations of
N supersymmetric Yang-Mills can be constructed through an SL(2,R) sequence of
T-dualities and coordinate shifts. We apply this transformation to multicenter
solutions and derive supergravity backgrounds describing the Coulomb branch of
N=1 theories at strong 't Hooft coupling as marginal deformations of N=4
Yang-Mills. For concreteness we concentrate to cases with an SO(4)xSO(2)
symmetry preserved by continuous distributions of D3-branes on a disc and on a
three-dimensional spherical shell. We compute the expectation value of the
Wilson loop operator and confirm the Coulombic behaviour of the heavy
quark-antiquark potential in the conformal case. When the vev is turned on we
find situations where a complete screening of the potential arises, as well as
a confining regime where a linear or a logarithmic potential prevails depending
on the ratio of the quark-antiquark separation to the typical vev scale. The
spectra of massless excitations on these backgrounds are analyzed by turning
the associated differential equations into Schrodinger problems. We find
explicit solutions taking into account the entire tower of states related to
the reduction of type-IIB supergravity to five dimensions, and hence we go
beyond the s-wave approximation that has been considered before for the
undeformed case. Arbitrary values of the deformation parameter give rise to the
Heun differential equation and the related Inozemtsev integrable system, via a
non-standard trigonometric limit as we explicitly demonstrate.Comment: 43 pages, Latex, 2 figures. v2: References added. v3: small typos
corrected, published versio
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